130 Thermal History of the Universe
a constant, except at the thresholds where particle species decouple. The physical
meaning of entropy of a system is really its degrees of freedom multiplied by some
constant, as one sees here. In Equation (6.12) we saw that the entropy density can
also be written
푠≡푆
푉
=
3
2
휁( 3 )푔훾푁훾, (6.60)
where푁훾is the number density of photons. Between two decoupling thresholds we
then have
d푆
d푡= d
d푡(
2 푔∗
3
푉푎S푇^3
푘
)
= 0. (6.61)
The second law of thermodynamics requires that entropy should be conserved in
reversible processes, also at thresholds where푔∗changes. This is only possible if푇
also changes in such a way that푔∗푇^3 remains a constant. When a relativistic particle
becomes nonrelativistic and disappears, its entropy is shared between the particles
remaining in thermal contact, causing some slight slowdown in the cooling rate. Pho-
tons never become nonrelativistic; neither do the practically massless neutrinos, and
therefore they continue to share the entropy of the Universe, each species conserving
its entropy separately.
Let us now apply this argument to the situation when the positrons and most of
the electrons disappear by annihilation below 0.2MeV. We denote temperatures and
entropies just above this energy by a subscript ‘+’, and below it by ‘−’. Above this
energy, the particles in thermal equilibrium are훾,e−,e+. Then the entropy
푆=^2
3
(
2 +^7
2
)
푉푎S푇+^3
푘
. (6.62)
Below that energy, only photons contribute the factor푔∗=2. Consequently, the ratio
of entropies푆+and푆−is
푆+
푆−=^11
4
(
푇+
푇−
) 3
. (6.63)
But entropy must be conserved so this ratio must be unity. It then follows that
푇−=
(
11
4
) 1 ∕ 3
푇+= 1. 40 푇+. (6.64)
Thus the temperature푇훾of the photons increases by a factor of 1.40 as the Universe
cools below the threshold for electron–positron pair production. Actually, the tem-
perature increase is so small and so gradual that it only slows down the cooling rate
temporarily.
Neutrino Decoupling. When the neutrinos no longer obey the condition in Equa-
tion (6.57) theydecoupleorfreeze outfrom all interactions, and begin a free expan-
sion. The decoupling of휈휇and휈휏occurs at about 3.8MeV, whereas the휈edecouple at
2.3MeV. This can be depicted as a set of connecting baths containing different parti-
cles, and having valves which close at given temperatures (see Figure 6.3).